Uniform Poincaré inequalities on measured metric spaces

Abstract

Consider a proper geodesic metric space (X, d) equipped with a Borel measure μ. We establish a family of uniform Poincaré inequalities on (X,d,μ) if it satisfies a local Poincaré inequality Ploc, and a condition on the growth of volume. Consequently, if μ is doubling and supports Ploc then it satisfies a uniform (σ,β,σ)-Poincaré inequality. If (X,d,μ) is a Gromov-hyperbolic space, then using the volume comparison theorem in Besson et al. (Curvature-free Margulis lemma for Gromov-hyperbolic spaces, 2020), we obtain a uniform Poincaré inequality with the exponential growth of the Poincaré constant. Next, we relate the growth of Poincaré constants to the growth of discrete subgroups of isometries of X, which act on it properly. We show that if X is the universal cover of a compact CD(K,∞) space with K≤0, it supports a uniform Poincaré inequality, and the Poincaré constant depends on the growth of the fundamental group of the quotient space.